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Ellenberg
,
J. S. and A. Venkatesh.
(
2007
)
“Reﬂection Principles and Bounds for Class Group Torsion
,
”
International Mathematics Research Notices
,
Vol. 2007
,
Article ID rnm002
,
18 pages.
doi:10.1093/
imrn
/rnm002
Reﬂection Principles and Bounds for Class Group Torsion
Jordan S. Ellenberg
1
and Akshay Venkatesh
2
1
Department of Mathematics
,
University of Wisconsin
,
Madison and
2
Courant Institute of Mathematical Sciences
,
New York University
,
New York
Correspondence to be sent to: Akshay Venkatesh
,
Courant Institute of Mathematical Sciences
,
New York University
,
New York. email: [email protected]
We introduce a new method to bound
±
torsion in class groups
,
combining analytic ideas
with reﬂection principles. This gives
,
in particular
,
new bounds for the 3torsion part
of class groups in quadratic
,
cubic and quartic number ﬁelds
,
as well as bounds for
certain families of higher degree ﬁelds and for higher
±
. Conditionally on GRH
,
we obtain
a nontrivial bound for
±
torsion in the class group of a general number ﬁeld.
1 Introduction
The goal of the present article is to exhibit some bounds on the
±
part of the class group
of a number ﬁeld which improve on the trivial bound provided by the order of the en
tire class group. As such
,
they represent evidence towards a conjecture that the
±
part of
the class group of a number ﬁeld
L
of ﬁxed degree grows more slowly than any power of
the discriminant of
L
. Such conjectures have been suggested by Duke
[
1
],
for CM ﬁelds by
Zhang
[
2
,
page 10
]
as the “
±
conjecture
,
” and in a stronger form by Brumer and Silverman
[
3
,
“Question
CL
(
±
,
d
)
”
]
Proposition 3.4 gives the bound

D

1
/
3
+
±
for the 3part of the class group of
Q
(
√
−
D
)
. This improves the known bounds of
[
4
]
and
[
5
]
and has several corollaries
(
cf.
[
4
,
Section 4
])
. In combination with the techniques of
[
4
]
one obtains that there are at
most
N
0
.
169
...
elliptic curves over
Q
of conductor
N
. More directly
,
it implies that there
are
±

D

1
/
3
+
±
cubic extensions of
Q
with discriminant
D
.
Received July 18
,
2006
;
Revised January 12
,
2007
;
Accepted January 16
,
2007
Communicated by Bjorn Poonen
See http://www.oxfordjournals.org/our
journals/imrn/for proper citation instructions.
c
²
The Author 2007. Published by Oxford University Press. All rights reserved.For permissions
,
please email: [email protected]
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J. S. Ellenberg and A. Venkatesh
Proposition 3.6 is our most general unconditional result on
±
torsion. A particu
lar case of Proposition 3.6 is a nontrivial bound on the 3torsion in even degree exten
sions of
Q
with large Galois group
;
but it also has consequences for
±>
3 and entails
,
e.g.
a nontrivial bound for the 5torsion part of the class group of any quadratic extension
of
Q
(
√
5
)
.Finally
,
in Corollary 3.7 we apply these results to show a nontrivial bound on
3torsion for cubic and quartic extensions of
Q
.
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